Question

How do you find the explicit formula for the following sequence 1, 1/2, 1/4, 1/8,...?

Answer 1

`n^(th)` term `1/2^(n-1)` and sum up to `n` terms is `2(1-1/2^n)`

Explanation:

The sequence `{1,1/2,1/4,1/8,..}` is a geometric series of the type `{a, a, ar^2, ar^3,....}`, in which `a` - the first term is `1` and ratio `r` between a term and its preceding term is `1/2`.

As the `n^(th)` term and sum up to `n` terms of the series `{a, a, ar^2, ar^3,....}` is `ar^(n-1)` and `(a(1-r^n))/(1-r)` (as `r<1` - in case `r>1` one can write it as `(a(r^n-1))/(r-1)`.

As such `n^(th)` term of the given series `{1,1/2,1/4,1/8,..}` is `1xx(1/2)^(n-1)` or `1/2^(n-1)`

and sum up to `n` terms of the series is `(1xx(1-(1/2)^n))/(1-1/2)` or

`((1-1/2^n))/(1/2)` or `2(1-1/2^n)`.

Note that when `n->oo`, `1/2^n->0` and hence sum of the series tends to `2`.